BAGHDAD, June 13 (Reuters) - Islamist rebel fighters captured two more Iraqi towns overnight in a relentless sweep south towards the capital Baghdad in a campaign to recreate a mediaeval caliphate carved out of fragmenting Iraq and Syria.

U.S. President Barack Obama threatened military strikes against the radical Islamic State of Iraq and the Levant on Thursday, highlighting the gravity of ISIL's threat to redraw borders in an oil-rich region with the risk of any new entity turning into a launch-pad for attacks on Western interests.

In the spreading chaos, Iraqi Kurdish forces seized control of Kirkuk - an oil hub just outside their autonomous enclave that they have long seen as their traditional capital - as Iraqi government troops abandoned posts in panic over ISIL's advance.

Thrusting further to the southeast after their lightning seizure of the major Iraqi city of Mosul in the far north and the late dictator Saddam Hussein's hometown of Tikrit, ISIL entered two towns in Diyala province bordering Iran.

Saadiyah and Jalawla had fallen to the Sunni Muslim insurgents after government troops fled their positions, along with several villages around the Himreen mountains that have long been a hideout for militants, security sources said .

The Iraqi army fired artillery shells at Saadiyah and Jalawla from the nearby town of Muqdadiya, sending dozens of families fleeing towards Khaniqin near the Iranian border.

Obama said on Thursday he was considering "all options" to support Iraq's Shi'ite Muslim-dominated central government that took full control when the U.S. occupation ended in 2011, eight years after the invasion that toppled Saddam.

"I don't rule out anything because we do have a stake in making sure that these jihadists are not getting a permanent foothold in either Iraq or Syria," Obama said at the White House, when asked whether he was contemplating air strikes.

"In our consultations with the Iraqis, there will be some short-term immediate things that need to be done militarily," he said. A U.S. defence official said the United States had been flying surveillance drones over Iraq to help it fight ISIL.

U.S. officials later said that U.S. ground forces would not return to Iraq.

INTERNATIONAL ALARM

But Obama said military action alone was no panacea against ISIL. He alluded to long-standing Western complaints that Shi'ite Prime Minister Nuri al-Maliki has done little to heal sectarian rifts that have left many of Iraq's minority Sunnis, cut out of power since Saddam's demise, aggrieved and keen for revenge.

"This should be also a wake-up call for the Iraqi government. There has to be a political component to this," Obama said.

U.S. Vice President Joe Biden assured Maliki by telephone that Washington was prepared to intensify and accelerate its security support. The White House had signalled on Wednesday it was looking to strengthen Iraqi forces rather than meet what one U.S. official said were past Iraqi requests for air strikes.

But fears of jihadist violence spreading may increase pressure for robust international action. French Foreign Minister Laurent Fabius said international powers "must deal with the situation".

In Mosul, ISIL staged a parade of American Humvee patrol vehicles seized from a collapsing Iraqi army in the two days since its fighters drove out of the desert and overran the city.

Giving a hint of their vision of a caliphate, ISIL published Sharia rules for the territory they have carved out in northern Iraq, including a ban on drugs, alcohol, cigarettes and an edict on women to wear only all-covering, shapeless clothing.

ISIL militants were reported to have executed soldiers and policemen after their seizure of some towns.

On Friday, ISIL said it was giving soldiers and policemen a "chance to repent ... For those asking who we are, we are the soldiers of Islam and have shouldered the responsibility to restore the glory of the Islamic Caliphate".

Residents near the border with Syria, where ISIL has exploited civil war to seize wide tracts of the country's northeast, saw its militants bull-dozing tracks through frontier sand berms - as a prelude to trying to revive a mediaeval entity straddling both modern states.

ISIL has battled rival rebel factions in Syria for months and occasionally taken on President Bashar al-Assad's forces.

But its fighters appear to have held back in Syria this week, especially in their eastern stronghold near the Iraqi border, while their Iraqi wing was making rapid military gains.

WEAPONS INTO SYRIA

ISIL's Syria branch is now bringing in weapons seized in Iraq from retreating government forces, according to Rami Abdulrahman, head of the Syrian Observatory for Human Rights monitoring group.

Matthew Henman, Head of IHS Jane's Terrorism and Insurgency Centre said in a report that ISIL's capture of Iraqi territory along the Syrian border will give the group greater freedom of movement of men and materiel across the two countries.

"Light and heavy weaponry, military vehicles, and money seized by ISIL during the capture of Mosul will be moved into desert area of eastern Syria, which ISIL has been using as a staging ground for attacks," he said.

At Baiji, near Kirkuk, ISIL fighters ringed Iraq's largest refinery, underlining the potential threat to the oil industry.

Further south, the fighters extended their advance to towns only about an hour's drive from Baghdad, where Shi'ite militia were mobilising for a potential replay of the ethnic and sectarian bloodbath of 2006 and 2007.

Security and police sources said Sunni militants now held parts of the town of Udhaim, 90 km (55 miles) north of Baghdad. "We are waiting for reinforcements and we are determined not to let them take control," said a police officer in Udhaim.

"We are afraid that terrorists are seeking to cut the main highway that links Baghdad to the north."

TARGET BAGHDAD

ISIL and its allies took control of Falluja at the start of the year. It lies just 50 km (30 miles) west of Maliki's office.

ISIL has set up military councils to run the towns they captured, residents said. "'Our final destination will be Baghdad, the decisive battle will be there' - that's what their leader kept repeating," said a regional tribal figure.

The senior U.N. official in Iraq assured the Security Council that Baghdad was in "no immediate danger". The council offered unanimous support to the government and condemned "terrorism".

As with the concurrent war in Syria, the conflict cuts across global alliances. The United States and Western and Gulf Arab allies back the mainly Sunni revolt against the Iranian-backed Syrian President Assad, but have had to watch as ISIL and other Islamists have come to dominate large parts of Syria.

Now the Shi'ite Islamic Republic of Iran, which in the 1980s fought Saddam for eight years at a time when the Sunni Iraqi leader enjoyed quiet U.S. support, may share an interest with the "Great Satan" Washington in bolstering mutual ally Maliki.

The global oil benchmark prices have jumped, as concerns mounted that the violence could disrupt supplies from a major OPEC exporter. Iraq's main oil export facilities are in the largely Shi'ite areas in the south and were "very, very safe", Oil Minister Abdul Kareem Luaibi said.

The million-strong Iraqi army, trained by the United States at a cost of nearly $25 billion, is hobbled by low morale and corruption. Its effectiveness is hurt by the perception in Sunni areas that it pursues the hostile interests of Shi'ites. (Additional reporting by Oliver Holmes in Beirut, Ziad al-Sinjary in Mosul Isabel Coles in Arbil, and Washington bureau; Writing by Mark Heinrich, editing by Peter Millership)